The Collaborative Research Centre (CRC) “Rough Analysis, Stochastic Dynamics and Related Fields” (SFB/Transregio 388) is one of eleven new Collaborative Research Centres that the German Research Foundation (DFG) will finance over the next four years. TU Berlin is the leading university; FU Berlin and HU Berlin are co-applicants. The spokesperson is MATH+ Distinguished Fellow Peter K. Friz (TU Berlin), who heads the Department of Mathematics, specifically Stochastics with a focus on Financial and Insurance Mathematics at TU Berlin, and conducts research at the Weierstrass Institute for Applied Analysis and Stochastics (WIAS). The spokesperson team also includes Nicolas Perkowski (FU Berlin) and Ulrich Horst (HU Berlin).

The DFG presented the topic of the new CRC as follows: The CRC “Rough Analysis, Stochastic Dynamics and Related Fields” aims to analyze phenomena subject to random influences. To this end, the CRC takes fundamental scientific approaches and considers application examples from fields such as financial mathematics. The participating mathematicians combine rough analysis and stochastic dynamics, linking them with other related subfields of mathematics such as geometry and algebra. Central to this is the “theory of rough paths,” which is not only mathematically significant but also helps develop essential new approaches for modeling dynamic processes in the natural, engineering, economic, and social sciences.

THE THEORY OF ROUGH PATHS

Peter K. Friz elaborates: “We will investigate the interplay of rough analysis and stochastic dynamics. Stochastic dynamics builds on probability theory and stochastic analysis to study the development of systems under the influence of randomness. Over decades, it has profoundly impacted many fields such as statistical physics, financial mathematics, uncertainty quantification, quantum field theory, mathematical biology, and economics. Rough analysis, on the other hand, represents recent breakthroughs in mathematics, rooted in the theory of rough paths.” He further comments, “The theory of rough paths is central to my work. It is a theory of objects that come with a time parameter, modeling, for example, a particle moving over time. The motion path of this particle is a so-called rough path, akin to the jagged curve of a stock price.”

Initially motivated by the need to introduce robustness into noise/signals, rough analysis offers a nonlinear extension of distribution theory, crucial for understanding singular stochastic dynamics and capturing nonlinear effects of signals.

SIGNIFICANT CONTRIBUTIONS FROM BERLIN’S SCIENTISTS

Beyond its origins, rough analysis has recently developed profound mathematical structures with significant geometric and algebraic components. This is also evidenced by its inclusion in the prestigious “Mathematics Subject Classification,” a kind of map of mathematics. Scientists from Berlin have made significant contributions to this important development. They were supported by the cross-institutional and interdisciplinary excellence cluster MATH+, the Berlin-Oxford-DFG Graduate School “Stochastic Analysis in Interaction,” the European Research Council (ERC), the Einstein Foundation Berlin, and a previous DFG research group.

The overarching goal of the new CRC/Transregio is to investigate the influence of rough analysis on fields such as analysis, algebra/geometry, and probability theory, as well as closely related applied topics such as statistics, stochastic control theory, and mathematical finance. This also includes questions about differential equations under the influence of highly irregular random fluctuations, which also occur in statistical physics. New algebraic structures play an important role, which in turn influence the field of statistics. Last but not least, the use of rough structures in stochastic control theory and financial mathematics (keyword “rough volatility”) is also on the research agenda.

Cooperation partners include, among others, the Max Planck Institute for Mathematics in the Sciences in Leipzig (MPI MiS), Universität Potsdam, and Universität Konstanz.

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DFG Collaborative Research Centre:

Collaborative Research Centres are long-term university-based research institutions, established for periods of up to 12 years. These centres bring together researchers to collaborate on multidisciplinary research programs. They enable researchers to tackle innovative, challenging, complex and long-term research projects by coordinating and concentrating resources and expertise within the applicant universities. This setup facilitates the development of institutional priority areas and structural advancements. Cooperations with non-university research institutions are strongly encouraged. Collaborative Research Centres consist of numerous projects, the number and scope of which depend on the research program. Individual projects are led by one researcher or or jointly by several researchers.

LINKS

• DFG Press Release on the 11 new CRC (in German)
• TU Berlin Press Release on the SFB/Transregio 388 in Mathematics (in German)
• Peter Karl Friz at TU Berlin, at WIAS, and at Einstein Foundation Berlin